When a bond trades for more than par, then it is selling at a premium , which will pay a lower yield than its stated coupon rate, and when it is selling for less, it is selling at a discount , paying a higher yield than its coupon rate. When interest rates rise, bond prices decline, and vice versa. Bond prices will also include accrued interest, which is the interest earned between coupon payment dates.

Clean bond prices are prices without accrued interest; dirty bond prices include accrued interest. A bond pays interest either periodically or, in the case of zero coupon bonds, at maturity. Therefore, the value of the bond is equal to the sum of the present value of all future payments — hence, it is the present value of an annuity , which is a series of periodic payments. The present value is calculated using the prevailing market interest rate for the term and risk profile of the bond, which may be more or less than the coupon rate. For a coupon bond that pays interest periodically, its value can be calculated thus:.

What is the present value of the payments? The following table shows the amount received each year and the present value of that amount. As you can see, the sum of the present value of each payment equals the par value of the bond. Note that the above formula is sometimes written with both C and r divided by 2; the results are the same, since it is a ratio.

Then, since there are 10 semiannual payment periods , the market interest rate is divided by 2 to account for the shorter period:. Case 2: 1 Annual Coupon Payment, resulting in 5 payment periods at the market interest rate:. In the primary bond market , where the buyer buys the bond from the issuer, the bond usually sells for par value, which is equal to the bond's value using the coupon rate of the bond.

However, in the secondary bond market, bond price still depends on the bond's value, but the interest rate to calculate that value is determined by the market interest rate, which is reflected in the actual bids and offers for bonds. Additionally, the buyer of the bond will have to pay any accrued interest on top of the bond's price unless the bond is purchased on the day it pays interest.

## Dirty vs. Clean Muni Bond Pricing

When bond prices are listed, the convention is to list them as a percentage of par value, regardless of what the face value of the bond is, with being equal to par value. This pricing convention allows different bonds with different face values to be compared directly.

Thus, a point's actual value depends on the face value of the bond. No commission is charged when buying or selling bonds. A bond dealer makes money through the spread —the difference between the bid price , which is what the dealer is willing to pay for a bond, and the ask price , which is what the dealer is selling the bond for. Another reason for this convention is that a point is not equal to a dollar, but a decimal base would still be more convenient.

The pricing convention is to list the point after a dash. The integer point value, in this case , is known as the handle.

When traders negotiate, the handle is usually known and not expressed. Listed bond prices are clean prices aka flat prices , which do not include accrued interest. Most bonds pay interest semi-annually. For settlement dates when interest is paid, the bond price is equal to the flat price. Between payment dates, accrued interest must be added to the flat price, which is often called the dirty price aka all-in price , gross price :. Accrued interest is the interest that has been earned, but not paid, and is calculated by the following formula:.

When you actually buy a bond on the secondary market, you would have to pay the former owner of the bond the accrued interest. If this were not so, you could make a fortune buying bonds right before they paid interest then selling them afterward.

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Because the interest accrues every day, the bond price increases accordingly until the interest payment date, when it drops to its flat price, then starts accruing interest again. In calculating the accrued interest, the actual number of days was counted from the last interest payment to the value date. This is because clean prices are more stable over time than dirty prices. Of course, when a bond is bought or sold in the secondary market, it is the dirty price that is paid.

Because bonds are quoted as clean prices, dirty prices must be calculated by ascribing a value to accrued interest and adding it to the clean price. Conceptually, we should add the market value of accrued interest to the clean price. Accordingly, there are conventions for ascribing value to accrued interest. To calculate the numerator of [2], you calculate the number of days between the last coupon date and the settlement date. The manner in which you do so depends on the applicable convention.

Calculation of the denominator also depends on the applicable convention.

## Bond Accrued Interest

If a bond trades on or after the ex-coupon date, the seller keeps the coupon. However, the buyer will own the bond during a small fraction of the coupon period. The seller must pay him the interest that accrues during that brief period. This is sometimes called negative accrued interest, because the interest is subtracted from the invoice price. Formula [2] is not used.